Sixth Grade: Mathematics Model Lesson for Unit #2: Meaning of Fractions, Decimals, Ratios, and Percents

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1 Meaning f Fractins, Decimals, Ratis, and Percents Sixth Grade: Mathematics Mdel Lessn fr Unit #2: Meaning f Fractins, Decimals, Ratis, and Percents Overarching Questin: Hw can reasning with ratis and rates help slve real wrld and mathematical prblems? Previus Unit: Factrs and Multiples This Unit: Meaning f Fractins, Decimals, Ratis, and Percents Next Unit: Tw-Dimensinal Gemetry Questins t Fcus Assessment and Instructin: 1. Hw are ratis and decimals alike? Hw are they different? 2. Hw can yu determine if tw ratis are equivalent? Intellectual Prcesses (Standards fr Mathematical Practice): Mdel with mathematics: Slve prblems invlving cmparisns f amunts and measurements. Key Cncepts: fractin percent decimal equivalence f fractins and ratis rati This dcument is the prperty f MAISA. Page 1 f 5

2 Meaning f Fractins, Decimals, Ratis, and Percents Lessn Abstract This lessn is an applicatin f equivalent fractins. Having a cncept f equivalent fractins is imprtant fr simplifying fractins. The number-sense f recgnizing equivalent fractins is useful when students study slpe and prprtins. Pairs (r grups f) students use a cup f beans t find ratis t express the number f marked beans in the cup cmpared t the ttal number f beans in the cup. Theretically, each sampling rati shuld be essentially the same. The decimal representatin f each rati cnfirms that the ratis are, indeed, apprximately equivalent. Cmmn Cre State Standards Ratis and Prprtinal Relatinships (6.RP) Understand rati cncepts and use rati reasning t slve prblems. 1. Understand the cncept f a rati and use rati language t describe a rati relatinship between tw quantities. Fr example, The rati f wings t beaks in the bird huse at the z was 2:1, because fr every 2 wings there was 1 beak. Fr every vte candidate A received, candidate C received nearly three vtes. 2. Understand the cncept f a unit rate a/b assciated with a rati a: b with b 0, and use rate language in the cntext f a rati relatinship. Fr example, This recipe has a rati f 3 cups f flur t 4 cups f sugar, s there is 3/4 cup f flur fr each cup f sugar. We paid $75 fr 15 hamburgers, which is a rate f $5 per hamburger. (Expectatins fr unit rates in this grade are limited t nn-cmplex fractins.) 3. Use rati and rate reasning t slve real-wrld and mathematical prblems, e.g., by reasning abut tables f equivalent ratis, tape diagrams, duble number line diagrams, r equatins. a. Make tables f equivalent ratis relating quantities with whle number measurements, find missing values in the tables, and plt the pairs f values n the crdinate plane. Use tables t cmpare ratis. c. Find a percent f a quantity as a rate per 100 (e.g., 30% f a quantity means 30/100 times the quantity); slve prblems invlving finding the whle, given a part and the percent. The Number System (6.NS) Apply and extend previus understandings f numbers t the system f ratinal numbers. 6. Understand a ratinal number as a pint n the number line. Extend number line diagrams and crdinate axes familiar frm previus grades t represent pints n the line and in the plane with negative number crdinates. *d. Develp an understanding f fractins, decimals, and percents and the relatinships between and amng the cncepts and their representatins. *e. Understand and use equivalent fractins. *f. Cmpare and rder fractins. This dcument is the prperty f MAISA. Page 2 f 5

3 Meaning f Fractins, Decimals, Ratis, and Percents *g. Mdel situatins invlving fractins, decimals and percents and use physical mdels (fractins strips, number line, partitin, grid area, percent bar mdels) and drawings t help reasn abut the situatins. * Indicates items added t CCSS Instructinal Resurces: Paper cups and tw clrs f dried beans (similar in size), Sequence f Lessn Activities Lessn Title: Bean Cunting and Ratis Selecting and Setting up a Mathematical Task: By the end f this lessn what d yu want yur students t understand, knw, and be able t d? In what ways des the task build n student s previus knwledge? Understand and use ratis t represent quantitative relatinships. Cnvert ratis t decimals and cmpare the results. Understand the cncept f a rati and use rati language t describe a rati relatinship between tw quantities. Use rati and rate reasning t slve real-wrld and mathematical prblems by reasning abut tables f equivalent ratis r decimal equivalents. In this case ratis f the dark clred beans t light clred beans in the sample will be used t set up ratis frm six different samples. The number f dark and light clred beans will be given t the students. Students have studied equivalent fractins, have the ability t interpret a fractin as divisin f the numeratr by the denminatr, and can cmpare tw decimals up t the thusandths place. Nw they will cmpare ratis frm samples f a bean ppulatin by use f equivalent fractins and decimal representatins f these fractins. If I have the rati 3/4, what are sme ratis that are equivalent? Ask students t explain hw they were able t arrive at the answer they have given. What questins will yu ask t help students access their prir knwledge? Hw can the rati 3/5 be changed int a decimal with three decimal places? Yu may see students use equivalent fractins t change 3 5 =? Other students may use divisin t generate the decimal equivalent. Launch: Hw will yu intrduce students t the activity s as t prvide access t all students while maintaining the cgnitive demands f the task? Prepare a cup f 40 beans in which 20 f the beans are a dark clr and the ther 20 are a light clr. It is imprtant that bth sets f beans are similar in size. This will be used as the demnstratin set. If I scp ut 14 f the beans, hw many wuld I expect t be dark? Hw culd this result be written as a rati? Hw can yu cmpare this rati with the rati representing the dark clred beans t all f the beans in the cup? Have students wrk individually t respnd and then share their answers with anther student. If students all used equivalent fractins, then discuss ther ways t represent this relatinship (i.e. decimals). If sme students already used decimals, This dcument is the prperty f MAISA. Page 3 f 5

4 Meaning f Fractins, Decimals, Ratis, and Percents have them share their ideas with the rest f the class. Using the same demnstratin cup f beans, have students cnsider hw many beans will be dark if I scp ut a sample f 27 beans. Since this will nt be an bvius answer (13.5), it allws students t cnsider whether a decimal answer makes sense given the cntext. Fr the class activity, prepare a cup with 45 beans fr each pair f students. Select 30 dark clred beans and 15 light clred beans. Activity Sheet A is prvided t recrd the results f six samples the students will take. All students can d the same rati f dark beans t the whle sample. Check t be sure students knw that all beans must be returned t the cup and the cup shaken befre a new sample is selected. What will be heard that indicates that the students understd what the task is asking them t d? Supprting Student s Explratin f the Task: Students will be writing ratis f dark beansinsample and cmparing them t ttal # f beans the knwn rati in the cups given. The cmparisns shuld be made by using either equivalent fractins r decimals. A calculatr may be used t supprt these cmputatins. What questins will be asked t fcus students thinking n the key mathematics ideas? What questins will be asked t assess student s understanding f key mathematics ideas? What questins will be asked t encurage all students t share their thinking with thers r t assess their understanding f their peer s ideas? Hw will yu extend the task t prvide additinal challenge? What cnnectins wuld yu expect t see in the ratis f dark beans t ttal beans fr the different samples that yu select? Hw will these ratis be related t the dark beans t the ttal beans in yur ppulatin (cupful) f beans? Hw des the average rati f the decimals fr all six trials cmpare t the decimal yu calculated fr the whle cup? Why d yu think this relatinship exisits? Hw are ratis similar t and different frm fractins? What methd did yu use t cmpare yur ratis? Is there any ther way t cmpare the ratis? Hw are the strategies yu used similar t r different frm yur past wrk with fractins? Hw can yu use the infrmatin yu knw abut the riginal cup f beans and it s dark beansinsample rati t cmpute the missing values in questins ttal # f beans 2 and 3 n the wrksheet? Prvide a variety f different cups with simpler r mre cmplicated ratis f dark beans : ttal beans geared t the levels f yur students. If students have already studied percents, have them cmpare the percentage f each sample that is dark, explaining hw t find the percent in each case. Give students a cup f beans, sme dark and sme light and tell them the ttal number f beans. Ask students t cme up with a plan t determine hw many f the beans in the cup are marked. This dcument is the prperty f MAISA. Page 4 f 5

5 Meaning f Fractins, Decimals, Ratis, and Percents Sharing and Discussing the Task: What specific questins will be asked s that all students will: Hw clse did each f the decimals yu calculated cme t the actual rati? What seems t be a reasnable range f values? Make sense f the mathematical ideas that yu wanted them t learn? Expand n, debate, and questin the slutins being shared? Make cnnectins between the different strategies that are presented? Lk fr patterns? Begin t frm generalizatins? Given the rati f dark beans t ttal beans as a decimal, hw culd yu find the rati f dark beans t ttal beans? Given the rati (written as a decimal) f dark beans t ttal beans, can yu find the ttal number f beans in the cup? What additinal infrmatin wuld yu need t d this calculatin? Given a pair f ratis, hw can yu tell if they are equivalent? What will be seen r heard that indicates all students understand the mathematical ideas yu intended them t learn? Frmative Assessment: Ask students t predict utcmes fr these examples: 1. In this class there are 28 students. One half f them are female. Hw many f them d yu expect t be female? If 9 students are called t the ffice, hw many are likely t be girls? [4 t 5] 2. A flrist used 25 flwers t make a buquet. Five f the flwers are daffdils. If 10 flwers are remved frm the bunch at randm, hw many f them d yu expect t be daffdils? 3. Write anther example f yur wn. This dcument is the prperty f MAISA. Page 5 f 5